Sobolev Space dual

Question

I'm interested in the dual space of the Sobolev space $H^1(\Omega)$ for $\Omega$ a bounded smooth domain. Of course, because $H^1(\Omega)$ being a Hilbert space, it's dual is isomorphic to itself, but this is if we use as dual pairing the $H_1(\Omega)$ scalar product: $$(u,v)_1 = \int_\Omega \nabla u \nabla v + \int_\Omega u v$$ but what I want is a representation that is compatible with the $L^2(\Omega)$ duality product, similarly as what happen with the space $H_0^1(\Omega)$, that is also isomorphic to it's dual, but with the $L^2(\Omega)$ product, is represented as derivatives of $L^2(\Omega)$ functions. Moreover, because of the inclusions $$ H_0^1(\Omega) \subseteq H^1(\Omega) \subseteq L^2(\Omega)$$ then we have $$ L^2(\Omega) \subseteq (H^1(\Omega))^* \subseteq H^{-1}(\Omega)$$ so the representation should be the functions of the for $f + \sum \partial_i g_i$ for $f, g_i\in L^2(\Omega)$ that satisfy certain additional condition.